As an infinitesimal, dx in this expression is always either only positive, or only negative . And never 0. But smaller than any real (bigger if negative) . That is the definition of infinitesimal in non-standard analysis-in hyperreal set. The theory that works in calculus without using limits, so I guess application of limits in this case is misleading , but I do not know what to apply.

dx in this case is not a limit dx->0 as real, but an extension of real set, or, rather set in which reals are just subset. It is a different type of number, so to say- infinitesimal. So is 1/dx = 1/infinitesimal= definition of infinity (also belongs to hyperreals).